Crystal fields, exchange and dipolar interactions, and noncollinear magnons of erbium oxide

We simulate the properties of magnons in erbium oxide, a noncollinear antiferromagnet, from an effective single-ion Hamiltonian, including exchange and long-range dipolar interactions. We parametrize the crystal field splitting of Er$_2$O$_3$ using Steven's operators and obtain the effective symmetry-dependent exchange constants between different erbium ions quenched by the crystal field at different symmetry sites. We apply the Holstein-Primakoff transformation to the noncollinear spin system and employ paraunitary diagonalization for the effective spin Hamiltonian. The addition of the dipolar interaction to the exchange magnon dispersion changes the magnon bands drastically. The long-range nature of the dipolar interaction provides challenges to convergence, however we find that the averaged and normalized difference in the magnon dispersion is less than an averaged factor of $10^{-6}$ if the dipolar interaction is included out to the fortieth nearest neighbor.